60 problems

In the setting of Theorem, let $\bm{l}=\bm{l}(\beta)$ be the length vector associated with the construction of $\operatorname{AF_\beta}$ gravitational instantons. Uniqueness conjec…

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let…

Direction-of-approach conjecture. The surface parts of the limit image should depend only on the boundary point of moduli space approached, whereas the complex of straight line seg…

Uniqueness conjecture. It may be possible that there is only one boundary point of the compactified moduli space having a finite-energy limit of map energies for the given set of m…

Hitchin–Goldman conjecture. The function $e$ has a unique minimum.

For the $n$-lump sector of the ${\mathbb{C}}{{P}}^1$ model on $S^2$, let $\tau^{eq}(\mu)$ be the equivariant coercivity constant for the map $\phi:z\mapsto \mu z^n$, with…

Harmonic extension conjecture. The admissible boundary map can be extended to a harmonic map from $X$ to $Y$ with bounded energy density if and only if the inverse image under the…

Let $f:\mathbb{S}^3\to\mathbb{S}^2$ be a harmonic map, where both spheres have their round metrics. Let $\pi:\mathbb{S}^3\to\mathbb{S}^2$ be the Hopf fibration. Eells' conjecture.…

Let $X \in \mathcal{T}^{\pm}(\mathbb{S}, \mathbb{M})$ and let $\hat{X}$ be as before. Write $X_n={\mathrm{PSL}_n(\mathbb{C})}/{\text{PSU}(n)}$ for the symmetric space and let…

Local Hölder regularity conjecture. Then $u$ is locally Hölder continuous in $\Omega$.

Extreme black hole uniqueness conjecture. At least one tangent map parameter must be nonzero:

Harmonic-map metastability conjecture. The relaxation time of the corresponding Langevin dynamics grows exponentially fast with $\beta$ if and only if $\mathcal E$ has non-trivial,…

Let $(a,b)$ satisfy $0\leq a\leq 1/2$ and $a^2+b^2\geq 1$, let $\mathbb{T}^2=\mathbb{R}^2/\Gamma_{a,b}$ be the corresponding flat torus with flat metric $g_{a,b}$, and let…

Let $(\mathbb S^n,g_{std})$ and $(\mathbb{S}^m,g_{std})$ be round spheres, and let $(\mathbb C \mathbb P ^n,g_{FS})$ and $(\mathbb C \mathbb P ^m,g_{FS})$ be complex projective spa…

Let $M$ and $N$ be hyperbolic surfaces, let $p>2$, and let $u_p:M\rightarrow N$ satisfy the $J_p$-Euler–Lagrange equations. Regularity conjecture. The map $u_p$ belongs to…

Let $M$ be a pinched Hadamard manifold, meaning a simply connected, complete Riemannian manifold whose sectional curvature is bounded between two negative constants. Let…

Weak Whitney theorems. The space of harmonic immersions in $\operatorname{\mathfrak{M}}$ is open and dense, and connected if $\dim M\geq 5$; moreover, the space of harmonic embeddi…

Let $S$ be a surface and let $G$ be a real Lie group of rank $2$. A $(\overline{\mathfrak{a}^{+}},k)$-mixed structure on $S$ is the structure whose dual $(\mathfrak{a},W)$-complex…

Nitsche-threshold conjecture. One has $r_\diamond<R$, and for every $r>r_\diamond$ there exists a minimizing $\rho$-harmonic diffeomorphism from $\mathbb{A}(r,1)$ onto…

Hélein's conjecture. There exist an orthonormal moving frame $(\mathbf e_1,\mathbf e_2)$ and a function $f$ such that

Let $M$ be the manifold under consideration, and let $u:M\to S^1$ be an $\infty$-harmonic map in a fixed homotopy class. Uniqueness conjecture. The map $u$ is unique up to rotation…

Let $\Sigma$ be a Riemann surface and let $\mathcal M_{\text{Higgs}}(\Sigma)$ be the moduli space of Higgs bundles, equipped with its Hitchin fibration. A Hitchin section is the se…

Let $\Omega$ be a bounded open domain in an $n$-dimensional Alexandrov space with curvature bounded from below by $k$, and let $N$ be a compact smooth Riemannian manifold. Suppose…

Let $D$ and $\Omega$ be doubly connected domains with smooth boundaries, and let $\rho$ be a smooth non-vanishing metric defined on the closure of $\Omega$. Write…

Let $M$ be a Riemannian manifold and let $N$ be a Riemannian manifold of negative sectional curvature. Consider the energy functional on a connected component of…